Convergence of multi-class systems of fixed possibly infinite sizes

TL;DR

This paper studies the convergence behavior of multi-class systems with both finite and infinite classes under partial exchangeability, revealing their sampling structure and establishing conditions for convergence of such systems.

Contribution

It characterizes the conditional law of multi-class systems with finite and infinite classes and links their convergence to the convergence of associated measure vectors.

Findings

Conditional law corresponds to independent sampling within classes

Convergence of systems is equivalent to convergence of measure vectors

Provides a framework for analyzing multi-class systems with infinite classes

Abstract

Multi-class systems having possibly both finite and infinite classes are investigated under a natural partial exchangeability assumption. It is proved that the conditional law of such a system, given the vector of the empirical measures of its finite classes and directing measures of its infinite ones (given by the de Finetti Theorem), corresponds to sampling independently from each class, without replacement from the finite classes and i.i.d. from the directing measure for the infinite ones. The equivalence between the convergence of multi-exchangeable systems with fixed class sizes and the convergence of the corresponding vectors of measures is then established.